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I got the same problem.
Could we do this : forget the first part of the tutorial. Could we export our private keys from Metamask and send them to myetherwallet and then claim them like the rest of the guide tells us to? I have 0 experience with this, so I don't know if it's possible

Unless you haven´t claimed your tokens already via the interface I assume that there is no way to extract the PKs.
In the meantime I found this post from @xbm and it seems we are not the only ones having that problem. It seems that also Dan is aware and they are working on setting up additional RPC servers. So it seems to be a capacity issue. So I recommend folling the thread over there to keep up to date.

https://steemit.com/eos/@xbm/troubleshooting-eos-distribution-app-with-metamask